the Feynman-Kac formula
/ FY-nman KOTS /
There is a second, completely different way to read a diffusion equation — not as a smoothing of profiles, but as a story about random walkers. Diffusion is the macroscopic shadow of countless microscopic particles jiggling at random (Brownian motion). The Feynman-Kac formula makes this precise: it writes the solution of a parabolic PDE as an average over random paths. To find the temperature at a point, you imagine a particle doing Brownian motion backward from that point and you average the data it meets. The deterministic PDE and the random process are two languages for the same object.
Concretely, consider u_t = (1/2) Laplacian u - V(x) u with initial data u(x, 0) = g(x), where V is a 'potential' (a position-dependent killing or discount rate). The Feynman-Kac formula says u(x, t) = E[ g(B_t) exp( - integral from 0 to t of V(B_s) ds ) | B_0 = x ], where B_s is Brownian motion started at x and E is the expectation (average) over all its random paths. Read it piece by piece: B_s is the random walker; g(B_t) is the initial value sampled at wherever the walker is after time t; the exponential of minus the integral of V weights each path by how much 'potential' it accumulated along the way; and the expectation averages over the entire cloud of possible paths. With V = 0 this collapses to u(x, t) = E[g(B_t)], which is exactly convolution against the heat kernel — the Gaussian density of B_t IS the heat kernel.
Why this dual viewpoint earns its place at the parabolic summit: it converts analysis into probability and back. PDE facts become path facts — the maximum principle is just 'an average cannot exceed the largest value being averaged'; positivity preservation is automatic because you are averaging nonnegative data; infinite propagation speed mirrors the fact that Brownian paths can reach anywhere in any positive time. It also turns into a computational tool: Monte Carlo simulation of random paths estimates PDE solutions in high dimensions where grids fail, and it is the mathematical backbone of option pricing, where the Black-Scholes PDE is solved as an expected discounted payoff.
Pure heat equation u_t = (1/2) Laplacian u, u(., 0) = g. Feynman-Kac (with V = 0) gives u(x, t) = E[g(B_t) | B_0 = x] = integral of (2 pi t)^{-n/2} e^{-|x-y|^2/(2t)} g(y) dy. The expectation over Brownian paths is literally the convolution of g with the heat kernel — probability and the heat kernel are the same statement.
The solution at a point is the average of the data over random (Brownian) paths from that point.
Feynman-Kac is a representation of parabolic (and elliptic) solutions, not of hyperbolic ones — there is no honest probabilistic path-average for the wave equation, because the underlying process is diffusion, which has the wrong (infinite-speed, smoothing) character for waves. Despite the name's nod to Feynman's path integral for quantum mechanics, the rigorous formula lives on the diffusion side; the genuinely oscillatory Schrodinger case is only formal.